I'm wondering why the T2 is better than the BH. I fully understand both schematics, although I'm not at a level to say that one particular aspect will make or break the sound quality.

As far as I'm seeing it, the only significant difference is that the T2 has an extra +-100V of swing, a servo, and a different input stage. There are a bunch of other subtle changes, and like I said, I don't have the experience to say which subtle changes are critical. The output stage is effectively the same. Also, given that the output stage is the same, I definitely don't understand how the T2 is "more powerful" as people seem to say.

Since you fully understand both circuits and I do not, would you please explain me the function of the T2 active battery and how such a function is accomplished in BH?

Sure... the input stage to the T2 converts to the voltage domain earlier. The anodes of U2 have a voltage output from the input stage, which is extracted through the Q4/Q5 followers. This signal needs a voltage drop to get down to the desired gate potential of the Q26/Q27 gain stage. There is no good way to do this other than to put a DC voltage (a battery, if you will) in series with the output from the Q4/Q5 followers. The battery adds to the signal dropping the potential down to the necessary ~-500V.

The BH on the other hand keeps the signals in the current domain throughout this stage. The input stage JFETS draw current away from the 2SA970 current sources, and the remaining current which holds the input signal flows through the 2SA1156 common-base stage, which is still carrying a current-domain signal and therefore will drop whatever voltage it needs to in order to get the current down to the VBE multiplier, which then turns the current into a voltage when it's already at the ~-400V potential to feed the FET gain stages.

So, the difference is basically that the battery is not necessary in the BH because the voltage drop from input potential to cathode potential is done in the current domain, while in the T2 it is done in the voltage domain.

But, like I said, what I don't have is the technical knowledge or intuition to say why one is better than the other which is what I'm really curious about (as well as all the other differences in the topologies).

Can't offer much here ... except that I personally listened to KG's T2 with his SR-007's at the Chicago CanJam. The box was bigger than my color laser printer. He told me that he'd probably never make another one - there were just too many parts and too much complexity to make it worth it to build another one.